Theoretical framework for a decision support system for micro-enterprise supermarket investment risk assessment using novel picture fuzzy hypersoft graph

Risk evaluation has always been of great interest for individuals wanting to invest in various businesses, especially in the marketing and product sale centres. A finely detailed evaluation of the risk factor can lead to better returns in terms of investment in a particular business. Considering this idea, this paper aims to evaluate the risk factor of investing in different nature of products in a supermarket for a better proportioning of investment based on the product’s sales. This is achieved using novel Picture fuzzy Hypersoft Graphs. Picture Fuzzy Hypersoft set (PFHSs) is employed in this technique, a hybrid structure of Picture Fuzzy set and Hypersoft Set. These structures work best for evaluating uncertainty using membership, non-membership, neutral, and multi-argument functions, making them ideal for Risk Evaluation studies. Also, the concept of the PFHS graph with the help of the PFHS set is introduced with some operations like the cartesian product, composition, union, direct product, and lexicographic product. This method presented in the paper provides new insight into product sale risk analysis with a pictorial representation of its associated factors.


Introduction
The world is evolving with humanity learning from processing immense amounts of data available in various forms, no matter how convoluted it may seem, in the hope that it may lead to some insight. Processing the data in multiple ways has allowed for a more analytic and scenario-based understanding of current affairs, allowing for effective decision-making in science and business. Keeping this notion in mind, business individuals in product sales and managing supermarkets require product sales supply and demand analytics so that the stock is always the optimum amount to go off the shelves while in the best condition.
Utilizing this information allows for the design of promotional strategies for product sales benefitting the business. Most customers in supermarkets are not subjected to confer with the a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 prices while buying the product, so there is minimal need for them to confer with their internal reference prices, allowing marketing strategists more room to design marketing strategies for business growth. For this purpose, numerous mathematical and computational tools have been designed to analyze these marketing indicators. Among these computational methods, some methods are discussed, but among these methods, attributes are not divided into subattributes. To encounter this problem, this paper presents a novel concept of picture fuzzy hypersoft graphs that allows for graphical analysis of multi-criteria decision-making problems while discussing a particular application regarding investment division and risk analysis of products in a supermarket or micro-enterprise setting. The paper first explains some mathematical literature and techniques used for market risk analysis. The novelty of the picture fuzzy hypersoft graph is defined by applying division of investment and evaluating the risk of supermarket products by tackling it as a multi-criteria decision-making problem.
The concept of fuzzy graph was developed by defining a graphical relationship among fuzzy sets by Rosenfeld in 1975 [1]. With the introduction of this concept, fuzzy set extensions like intuitionistic fuzzy set (IFS) [2] were further improved to picture fuzzy set (PFS) [3]. This extension allowed for addressing opinion based data (i.e., yes, no, abstain, and refusal) by addition of a neutral membership function to the already existing IFS. The operators and basic definitions of the PFS concept have been presented in [4,5]. PFS has been discussed in numerous application based research in diverse fields like information technology, pharmacy, operational research, and business. The above-discussed fuzzy structures fall short when it comes to addressing data where the attributes are divided into multiple subattributes. For this purpose, Soft Set (SS) were extended to Hypersoft Set (HSS) by Smarandache which allowed modifying the one parameter set into a multi-parameter set in the domain set of function [6]. The properties of HSS were developed by Saeed et al. (i.e., HS subset, union, intersection, etc.) and developed hybrids of this HSS structure with structures from fuzzy and neutrosophic set theory [7,8]. Literature does revela some fuzzy hyperosft hybrid structures including Pythagorean fuzzy hypersoft set and Intuitionistic Fuzzy Hypersoft Set (IFHSS) which is a generalized version of the Intuitionistic Fuzzy Soft Set (IFSS) [9,10]. To elaborate the versatility and operation of the hybrid fuzzy hypersoft and neutrosophic hypersoft structures, they were applied for the development of medical decision support systems, pattern recognition to monitor the spread of COVID-19 with real-time population data over the period of 1.5 years [11][12][13][14][15][16][17][18].

Literature review
The development of a fuzzy graph structure lead to the expansion of hybrid structures of Fuzzy Sets and Graph Theory (i.e., balanced interval-valued fuzzy graphs [19], fuzzy planar graphs [20,21], m-step fuzzy competition graphs [22], fuzzy threshold graph [23], fuzzy cubic graph [24], and fuzzy k-competition graph [25]). The interval-valued fuzzy threshold graph was developed and its properties were explored by Pramanik et al. [26]. The concept of planarity was applied to a simple Bipolar Fuzzy Graph leading to an extension to the already existing structure to a more refined bipolar fuzzy planar graphs [27]. Some other hybrid extension of the Fuzzy Planar graphs include interval-valued fuzzy graph [28] and interval-valued fuzzy planar graph [29]. The concept of Fuzzy Hypergraphs was developed by Voskoglou et al. [30] alongside several other related graphical strctures. Another term called Intuitionistic fuzzy competition graph was introduced by Sahoo et al. [31] while Balanced intuitionistic fuzzy graphs were introduced by Karunambigai et al. [32]. When talking about Intuitionistic Fuzzy Graphs, Sahoo et al. applied these intricate structure to numerous real world applications [33][34][35].
Numerous studies related to Picture Fuzzy Graphs and their properties like edge domination have been reported in literature alongside their applications and hybrid structures [36][37][38]. The hybrid structures like the Picture Dombi Fuzzy Graph, interval-valued picture uncertain linguistic generalized Hamacher aggregation operators, and the q-rung picture fuzzy graph have significant applications in decision-making problems, design of decision support systems, and associating fuzzy graphical structures in topological spaces [39][40][41][42][43]. These decision support systems require fuzzy aggregation opertors that combine the observations recorded in the data for manipulation in a fuzzy environment [44]. Some examples of these studies include pattern recognition using novel similarity measures for T-spherical fuzzy sets, a policy design aid by using an averaging operator of T-spherical fuzzy set [45,46]. The Tspherical fuzzy set was presented as a generalization of the previously existing Fuzzy Set concepts (i.e., Intuitionistic fuzzy set and the Picture fuzzy set) [47].
Fuzzy set theory and concepts from Graph theory have been combined for the development of hybrid structures with better ability to handle and express data to extract better results. Picture fuzzy labeling graphs were reported by Devaraj et al. in 2020 alongside its application in decision making [47,48]. Another interesting concept of fuzzy cross-entropy for picture hesitant fuzzy sets and introduction of polarity in addressing of attributes using Bipolar soft sets were developed by Mahmood et al. for addressing attributes of different nature in decision making problems [49,50]. A generalized concept of picture fuzzy soft set alongside its application was put forward by Khan et al. [50,51]. Intuitionistic multiplicative set operational laws and their associated aggregation operators for processing was developed by Garg alongside its application in decision support systems [40]. Balanced Picture Fuzzy Graphs were explored in 2021 by Amanathulla et al. [51,52]. Numerous related hybrid fuzzy graph theoretical concepts relating to different aspects of their applications are presented in the following studies [53][54][55][56][57][58].
The concept of hybrids of hypersoft structures and graph theory have been discussed by Saeed et al. including intricate concepts like the hypersoft graph, a hypersoft subgraph, a complete hypersoft graph, and a hypersoft tree etc [59]. Due to the fast and evolving nature of businesses nowadays, its essential to calculate every single move with extreme delicacy and process a plethora of information in order to assess the risk with respect to inventory management. Literature reveals some studies that involve the use of mower based on fuzzy logic approach for effective and efficient methods for customer service [60]. Similar economic-mathematical models allow for the forecasting of growth of agricultural sector of Ukraine in order to make suitable assessment for people in business and agarians [61]. These assessments allow for timely making business decision based on facts and calculations reducing the risk in terms of loss. Another approach presented in [62] aims for evaluation of enterprise based systems with multiple criteria for analysis and designing a decision-support system. An approach presented in [63] applied clique covering of a fuzzy graph for parametric optimization for development of optimal business strategies.

Motivation
Numerous methods have been reported in the literature that has been used to solve MADM problems by using concepts from Fuzzy and Soft Sets. All these methods have some restrictions, such as when elements of the attributes set contain further sub-attributes and elements of each attribute set have no relationship. In order to address these difficulties, the concept of Picture Fuzzy Hypersoft graph (PFHSG) is introduced in this paper which is a hybrid structure of the Picture Fuzzy Hypersoft set with Graph Theory. The presented structure was then employed to develop a decision-support algorithm that uses a combination of concepts form the Hypersoft and Fuzzy Set theory. For a proper explanation, some basic terminologies are defined in Section 1, section 2 focuses on the presentation of properties of the PFHSG, while section 3 highlights the multi-attribute decision support system that was designed to based on the PFHSG for addressing intricate decision-making issues in a graphical manner. In Section 4, the algorithm is numerically applied on a risk assessment problem in order to elaborate its working principle and operations. The final section 4 is presents a brief conclusion of the paper.
Atanassov [2] defined IFS a direct extension of the fuzzy set which consist a membership function and a non-membership function. It overcomes defects of fuzzy sets.
Definition 1: Intuisionistic Fuzzy Set [2] An intuitionistic fuzzy set X on universe of discourse X = {x 1 , x 2 , . . .‥, x n } is an object of the form:L [4] In 2013, Coung [4] introduced PFS in order to solve inconsistent and uncertain information in real life. Picture fuzzy set consists of three functions: positive membership function, neutral membership function, and negative membership function. A good example of PFS is electoral voting. A Picture fuzzy set on universe of discourse X = {x 1 , x 2 , . . .‥, x n } is an object of the form:L

Definition 2: Picture Fuzzy Set
The set of all picture fuzzy subsets on universe of discourseL 1 is denoted by PFSs(X).
Some basic operation of PFS are defined as follows: Xg be two PFSs on universe X. Then the operations between the setsL 1 andL 2 are defined as follows: A new scientific instrument in which parameter space is finite or infinite, even if universal set is finite, is known as a soft set. It was proposed by Molodtsov [64] for parameterized uncertainty handling purposes. A mapping F : A ! PðUÞ (F, A) is called a soft set over U, where A is set of parameters. Definition 5: Picture Fuzzy Soft Set [65] In 2015, Yang et al [65] proposed PFSs which is combination of PFS and SS. Let E be a parametric space with a U universal set. Now, the set of all picture fuzzy sets be represented by U. In this scenario, a picture fuzzy soft set (PFSS) is a pair (F, A) where A � E and F is a mapping as shown: F : A ! PFðUÞ: Definition 6: Hypersoft Set [6] In 2018, Smarandache [6] put forward the concept of Hypersoft Set, which is a generalization of SS by transforming the mapping into a multi-attribute. Suppose b 1 , b 2 , . . .. . ., b n , for b � 1, be n distinct traits, whose corresponding trait values are respectively the sets Definition 6: Fuzzy Graph [35] A fuzzy graph G is defined by ordered paired of functions of μ and ρ, and G = (μ, ρ). μ represents a fuzzy subset of V (A finite non-empty set of vertices) and ρ represents a symmetric fuzzy relation on μ, i.e., μ : rðp; qÞ � minðmðpÞ; rðqÞÞ; 8p; q 2 V: Definition 7: Complete Fuzzy Graph [35] A fuzzy graph G = (μ, ρ) is called complete fuzzy graph if rðp; qÞ ¼ minðmðpÞ; rðqÞÞ; 8p; q 2 V: Definition 8: Strong Fuzzy Graph [35] A fuzzy graph G = (μ, ρ) is called strong fuzzy graph if rðp; qÞ ¼ minðmðpÞ; rðqÞÞ; 8p; q 2 V: Definition 9: Complement of a Fuzzy Graph [35] The complement of a fuzzy graph G = (μ, ρ) is a fuzzy graph and it is represented as G c = (μ c , ρ c ), where μ c = μ and rðp; qÞ ¼ minðmðpÞ; rðqÞÞ À rðp; qÞ; 8p; q 2 V: Definition 10: Picture Fuzzy Graph [35] Suppose P ¼ ðV; EÞ be a graph.

Picture fuzzy hypersoft graph
In [65], hybrid model of a PFS and a SS is defined. Keeping that notion in mind, we introduce PFHSS as an extension of PFSS, which helps for playing a crucial role in decision-making for multi-attribute characteristics.

Definition 11: Picture Fuzzy Hypersoft Set
Suppose m disjoint attribute-valued sets are p 1 , p 2 , p 3 , . . ., p m then their corresponding m distinct attributes are P 1 ; P 2 ; P 3 ; ::::; P m ,respectively. And P ¼ P 1 � P 2 � P 3 � :::: � P m . A mapping is given by € H : P ! PFðUÞ then pair ð € H; PÞ represent PFHSS. Definition 12: Picture Fuzzy Hypersoft Graph Let There are a total of 9 outcomes but for simplicity, only three outcomes have been selected for the analysis.L The vertices set of PFHSG presented in Table 1.
ÞÞ are shown in Figs 1 and 2, respectively. Fig 3 represents the above presented structure with its edge set presented in Table 2.

Application of picture fuzzy hypersoft graphs in decision support analysis
PFHSS is a vital hybrid structure to deal with problems faced in the real world. Studying inconsistent, incomplete, indeterminate, and multi-argument facts is highly beneficial when discussing problems in medical science, engineering, and business studies. Due to its versatile nature and wide range of applications, PFHSS has become an interesting and inventive subject for researchers. So, PFHSG can provide solutions to these kinds of problems. Here, the PFHS graph is used to deal with MADMP, and an algorithm is proposed to outline the thought process that illustrates the model's working.

Algorithm:
Step 1: Calculate the impact coefficient between the attributes K by $ ij ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi g is the PFHS graph edge between nodes K for i, j = 1, 2, . . ., n, and $ ij ¼ $ ji .
Step 2: Find the attribute of the alternative � s i below: Step 3: Computation of the score functions using the follwoing equation: score ð� Step 4: Selection of the best alternative by ranking the alternatives.
Step 5: Reporting of results.

Product risk assessment based for optimal design of investment strategies
In this section, numerical examples for the PFHSG MADM problem with picture fuzzy and hypersoft information are used to present the application of the proposed algorithms. An owner of supermarket wants to invest money and buy a variety of products based on numerous factors illustrated in the figure below. Now, a smart investment would be to to buy the products that have highest optimal value in terms of Return on investment, environmental factors, and shelf life of the product etc. Based on these factors, we propose an algorithm based on PFHS graphical structure to find the risk in investment when buying each product while addressing each of the factors addressed in the figure. The problem is illustrated below: Pharmacy} are four measurable alternatives for a businessperson starting a new business. Let E = {a 1 , a 2 , a 3 , a 4 } be attributes set where each a i stands for Gross margin, Product monthly sales, Customer satisfactions and environmental impact analysis respectively, whose corresponding attribute values are {H 1 , H 2 , H 3 } respectively. In order for the businessperson to efficiently invest in these four alternative investment options, a risk analysis must be done that addresses all the factors listed below to generate the quickest return on investment while also considering factors like the market image of the store and quality of the products.
Let the factors being considered for the risk analysis assessment be: • There are one hundred and eight outcomes but for simplicity, only three outcomes are explored and risk assessment is done on these three outcomes. The edge set of the outcomes is presented in Table 3.
Step 1: Computation of the impact coefficient between the attributes K: Step 2: The attributes of the alternative � s i is calculated below:   Step 3: Computation of Final score functions for ranking: Hence ð� s 4 Þ is best choice because score ð� s 4 Þ � score ð� s 3 Þ � score ð� s 1 Þ � score ð� s 2 Þ. This provides a clear indication that the Pharmacy section of the supermarket is the best for investment as it has the lowest risk for loss of investment while considering the factors listed in Fig 19. Following the Pharmacy is the Dairy products section of the store, with the Bakery at the third position and the Meat section at the fourth. This example only describes a minimal example for an explanation but can address the sub-attributes of attributes in a graphical manner which are in most cases neglected (Other than Hypersoft Structures) due to the inability to address all the factors addressed in Table 4. This tool is great for obtaining a graphical depiction for development and solving a decision-making problem. It works great as a decision support system as it has potential applications in just about any field, from medical diagnostic systems to mechanical engineering problems.

Comparative analysis
A comparative analysis of the proposed structure is provided in Table 4. Previously available structures in literature can address some attributes using fuzzy parameters like membership  and non-membership functions, but they fall short when there are sub-attributes. These are then addressed using Hypersoft structures that can deal with such issues, and the proposed hybrid structure covers the whole spectrum of characteristics allowing for deep and thorough analysis.

Conclusion
Risk evaluation has always been of great interest for individuals wanting to invest in various businesses, especially in the marketing and product sale centres. A finely detailed evaluation of the risk factor can lead to better returns in terms of investment in a particular business. This presents a MADM problem requiring numerous factors to be addressed simultaneously. Graph theory is an essential tool for solving MADM problems in various kinds of hybrid structures like PFS, PFSS, HS, and PFHSS. Picture fuzzy hypersoft graph is a new concept in the theory of graphs. PFHS graph has been applied to solve problems containing consistent, inconsistent, and multi-argument information. This article introduces basic operations like union, composition, cartesian product, direct product, and joint of PFHS graph. We proposed an algorithm in which a relationship is created among attributes and alternatives in the decision process to obtain the desired result. The structure is then used to address a risk analysis problem for investment distribution for product buying in a supermarket. In the future, we can extend this structure in neutrosophic, spherical and T-spherical hypersoft sets while applying the structures to actual data for real-world applications.